Three containers $A$, $B$ and $C$ are having mixtures of milk and water in the ratio $1 : 5$, $3 : 5$ and $5 : 7$ respectively. If the capacities of the containers are in the ratio $5 : 4 : 5$, find the ratio of milk to water, if the mixtures of all the three containers are mixed together?

### Concept & Mixtures with Unequal Capacities When mixing containers with different capacities and internal ratios, combine the LCM method for equalizing the ratio sums with a multiplier that reflects the capacity ratio of the containers. ### Step-by-Step Solution * **Given:** - Container A ratio = $1 : 5$ (Sum = $6$) - Container B ratio = $3 : 5$ (Sum = $8$) - Container C ratio = $5 : 7$ (Sum = $12$) - Ratio of capacities A:B:C = $5 : 4 : 5$ * **Calculation:** 1. Find the LCM of the ratio sums ($6, 8, 12$), which is $24$. 2. To account for the capacities, multiply the LCM by the capacity ratio for each container: - Capacity of A = $24 \times 5 = 120$ units - Capacity of B = $24 \times 4 = 96$ units - Capacity of C = $24 \times 5 = 120$ units 3. Calculate the amount of milk and water in each container: - Container A: Milk = $\frac{1}{6} \times 120 = 20$; Water = $120 - 20 = 100$ - Container B: Milk = $\frac{3}{8} \times 96 = 36$; Water = $96 - 36 = 60$ - Container C: Milk = $\frac{5}{12} \times 120 = 50$; Water = $120 - 50 = 70$ 4. Total Milk = $20 + 36 + 50 = 106$ units 5. Total Water = $100 + 60 + 70 = 230$ units 6. The final ratio of milk to water is $106 : 230$. 7. Simplifying the ratio by dividing by $2$ yields $53 : 115$. ### Exam Strategy & Shortcut Equalize the base denominators first ($6, 8, 12 \rightarrow$ scale to $24$). A's fraction scaled: $4:20$. B's fraction scaled: $9:15$. C's fraction scaled: $10:14$. Now multiply each by their respective capacity weights ($5, 4, 5$). A: $(4 \times 5) : (20 \times 5) = 20 : 100$ B: $(9 \times 4) : (15 \times 4) = 36 : 60$ C: $(10 \times 5) : (14 \times 5) = 50 : 70$ Summing them gives $106 : 230 = 53 : 115$. ### Common Pitfall A frequent error is forgetting to apply the capacity weights ($5:4:5$) after finding the common base for the internal fractions, treating them as if they were of equal volume. ### Final Answer Therefore, the correct answer is **$53 : 115$**.